a stable Levy process
/ lev-EE /
A stable Lévy process is the Lévy process whose increments follow a stable law — the unique self-similar Lévy processes, scaling like X_{ct} = c^(1/alpha) X_t in law. They are the canonical heavy-tailed jump processes and the only nontrivial Lévy processes invariant under this kind of rescaling, which makes them the natural limits of normalized random-walk sums when the step law has heavy tails outside the domain of attraction of the normal.
The process is parametrized by a stability index alpha in (0, 2]; alpha = 2 is Brownian motion (Gaussian, the only stable law with finite variance), and 0 < alpha < 2 gives genuine jump processes with characteristic exponent psi(theta) = -c |theta|^alpha (1 - i beta sgn(theta) tan(pi alpha/2)) + i mu theta in the symmetric case reducing to psi(theta) = -c |theta|^alpha. The Lévy measure is nu(dx) = c_+ x^(-1-alpha) dx on x>0 and c_- |x|^(-1-alpha) dx on x<0, a pure power law: there is no Gaussian part for alpha < 2, only jumps. Self-similarity follows from the homogeneity of psi: e^(ct psi(theta)) = e^(t psi(c^(1/alpha) theta)). The skewness parameter beta in [-1, 1] tilts the jump rates; beta = 0 is symmetric, beta = 1 with alpha < 1 gives the one-sided stable subordinator.
Stable processes model phenomena with bursts and heavy tails — turbulent transport, financial crashes, anomalous diffusion — precisely where the Gaussian fails. The non-negotiable honesty: for alpha < 2 a stable law has INFINITE variance (the tails decay only like |x|^(-alpha)), and for alpha <= 1 it has infinite mean, so the classical CLT and LLN do NOT apply; this is the whole point of the generalized CLT. A frequent error is to compute or quote a variance for a non-Gaussian stable process — it does not exist.
The Cauchy process (alpha = 1, symmetric) has psi(theta) = -c|theta| and Lévy measure nu(dx) = c|x|^(-2) dx; its increment over time t is a Cauchy law with scale ct, which has no mean and no variance. Self-similarity X_{ct} = c X_t in law means doubling the time horizon doubles the typical excursion size, unlike Brownian motion where it grows like the square root.
Cauchy process: alpha = 1, heavy tails, no mean or variance.
For 0 < alpha < 2 the variance is infinite and for alpha <= 1 even the mean is infinite, so the classical CLT and LLN do not hold — only the generalized CLT does. The Gaussian case alpha = 2 is the unique stable law with finite variance.